April 03, 2026
Let \(X\) be an affine variety and \({\mathfrak h}\subset\mathop{\rm Lie}(\mathop{\rm Aut}(X))\) be a solvable Lie subalgebra generated by a finite collection of locally finite Lie subalgebras. The authors of [1] wondered whether \({\mathfrak h}\) is itself locally finite. Here we present some criteria for the local finiteness of \({\mathfrak h}\). Under a certain additional assumption, we answer this question in the affirmative in the particular case where \(X\) is the affine plane \({\mathbb{A}}^2\).
All the algebraic varieties in this paper are defined over an algebraically closed field \({\Bbbk}\) of characteristic zero. Let \(X\) be an affine variety over \({\Bbbk}\), and \({\mathfrak h}\) a Lie subalgebra of \(\mathop{\rm Lie}(\mathop{\rm Aut}(X))\). We say that \({\mathfrak h}\) is locally finite if every \(f\in{\mathcal{O}}(X)\) belongs to a finite-dimensional vector subspace of \({\mathcal{O}}(X)\) invariant under \({\mathfrak h}\), see, e.g., [2]. A derivation \(\partial\in\mathop{\rm Der}({\mathcal{O}}(X))\) is said to be locally finite if the Lie subalgebra \({\Bbbk}\partial\) is locally finite. Recall the following questions.
Questions 1 ([1]). Let \({\mathfrak h}=\langle {\mathfrak h}_1,\ldots,{\mathfrak h}_k\rangle_{\mathop{\rm Lie}}\) be a solvable Lie subalgebra of \(\mathop{\rm Lie}(\mathop{\rm Aut}(X))\) generated by the locally finite Lie subalgebras \({\mathfrak h}_i\subset \mathop{\rm Lie}(\mathop{\rm Aut}(X))\), \(i=1,\ldots,k\). Is it true that \({\mathfrak h}\) is locally finite?
The author knows of no example where \({\mathfrak h}\) as above were not locally finite.
Recall that a locally finite Lie subalgebra \({\mathfrak h}\subset\mathop{\rm Lie}(\mathop{\rm Aut}(X))\) is finite-dimensional, see [2]. Question 1 is related to the following one.
Questions 2 (see [2]). Let \(\{{\mathfrak h}_i\}_i\) be a family of locally finite Lie subalgebras of \(\mathop{\rm Der}({\mathcal{O}}(X))\). Is is true that the Lie subalgebra \({\mathfrak h}\subset\mathop{\rm Lie}(\mathop{\rm Aut}(X))\) generated by the \({\mathfrak h}_i\) is locally finite provided that it is finite-dimensional?
Here, we only consider the case of a solvable Lie algebra \({\mathfrak h}\). In Section 2, after recalling some results from [1] and [2], we give the following finite-dimensionality criterion for such a Lie algebra \({\mathfrak h}\).
Proposition 3 (see Proposition 17). Let \({\mathfrak h}\subset\mathop{\rm Lie}(\mathop{\rm Aut}(X))\) be a Lie subalgebra as in Question 1. Then \({\mathfrak h}\) is finite-dimensional if and only if the derived ideal \([{\mathfrak h},{\mathfrak h}]\) is nilpotent.
In Section 3 we show that, in the case where \(X={\mathbb{A}}^2\), the answer to Question 1 is affirmative under an additional assumption.
Theorem 4 (cf. Theorem 19). A solvable Lie subalgebra \[{\mathfrak h}=\langle \partial_1,\ldots,\partial_k\rangle_{\mathop{\rm Lie}} \subset\mathop{\rm Lie}(\mathop{\rm Aut}({\mathbb{A}}^2))\] generated by locally finite derivations \(\partial_1,\ldots,\partial_k\) is locally finite and triangulable, provided \({\mathfrak h}\) contains a locally finite derivation \(\partial\) that is not locally nilpotent.
We believe that an analogue of this theorem, with a similar proof, also remains valid for normal affine toric surfaces and more general affine toric varieties. On the other hand, we do not know the answer to Question 1 in the case where \(X={\mathbb{A}}^2\) and all locally finite derivations in \({\mathfrak h}\) are locally nilpotent. Nevertheless, we prove the following result in this direction.
Theorem 5. Let \({\mathfrak h}=\langle \partial_1,\ldots,\partial_k\rangle_{\mathop{\rm Lie}}\), where the \(\partial_i\in\mathop{\rm Vec}^{0}({\mathbb{A}}^2)\) are nonzero locally nilpotent derivations. Suppose that the center of \({\mathfrak h}\) is nontrivial. Then \({\mathfrak h}\) is locally finite and triangulable.
Since the center of a nilpotent Lie algebra is nonzero, we can deduce from Theorems 4 and 5 the following corollary.
Corollary 6. For \(X={\mathbb{A}}^2\), the answer to Question 1 is affirmative provided that \({\mathfrak h}\) in the question is nilpotent.
The content of the paper is as follows. In Section 2 we consider Question 1 for general affine varieties, and we perform several successive reductions. In particular, we prove Proposition 3. In Section 3 we turn to the case of the affine plane. After recalling necessary preliminaries on the structure of \(\mathop{\rm Lie}(\mathop{\rm Aut}({\mathbb{A}}^2))\) as a bigraded Lie algebra, we prove Theorem 4. In Lemma 39 we treat the case, where \({\mathfrak h}\) is of rank \(2\), and in Proposition 41 the case, where \({\mathfrak h}\) is of rank \(1\). Besides, we show that every solvable subalgebra \({\mathfrak h}\subset\mathop{\rm Lie}(\mathop{\rm Aut}({\mathbb{A}}^2))\) as in Question 1 is triangulable, see Lemma 22. The latter does not hold, in general, for solvable locally finite Lie subalgebras of \(\mathop{\rm Lie}(\mathop{\rm Aut}({\mathbb{A}}^n))\) if \(n\ge 3\), see Remark 23.2. In subsection 3.6 we prove Theorem 5. Answering a question proposed by A. Regeta1, we show that a solvable subalgebra \({\mathfrak h}\subset\mathop{\rm Lie}(\mathop{\rm Aut}({\mathbb{A}}^2))\) consisting of locally finite derivations is filtered by a sequence of locally finite Lie subalgebras and is triangulable, see Corollary 42.
We can consider that every \({\mathfrak h}_i\) in Question 1 is one-dimensional. Indeed, being locally finite, \({\mathfrak h}_i\) is finite-dimensional and consists of locally finite derivations. Thus, every \({\mathfrak h}_i\) is generated by a finite set of locally finite derivations, and the same is true for \({\mathfrak h}\).
Recall that every locally finite derivation \(\partial\in\mathop{\rm Lie}(\mathop{\rm Aut}(X))\) admits a unique Jordan decomposition \(\partial=\partial_{\mathrm s}+\partial_{\mathrm n}\), where \(\partial_{\mathrm s}\) is semisimple and \(\partial_{\mathrm n}\) is locally nilpotent, they commute, and every vector subspace \(V\subset{\mathcal{O}}(X)\) invariant under \(\partial\) also is invariant under \(\partial_{\mathrm s}\) and \(\partial_{\mathrm n}\), see, e.g., [3]. Furthermore, \(\mathop{\rm ad}_{\partial}=\mathop{\rm ad}_{\partial_{\mathrm s}} +\mathop{\rm ad}_{\partial_{\mathrm n}}\) is a Jordan decomposition of \(\mathop{\rm ad}_{\partial}\) acting on \(\mathop{\rm Lie}(\mathop{\rm Aut}(X))\) (see [2]), and any vector subspace \(V\subset \mathop{\rm Lie}(\mathop{\rm Aut}(X))\) invariant under \(\mathop{\rm ad}_{\partial}\) also is invariant under \(\mathop{\rm ad}_{\partial_{\mathrm s}}\) and \(\mathop{\rm ad}_{\partial_{\mathrm n}}\), see [ibid].
These observations lead to the following lemma.
Lemma 7. Let \({\mathfrak h}\subset\mathop{\rm Lie}(\mathop{\rm Aut}(X))\) be a solvable Lie subalgebra of derived length \(l\) and \(\partial\in{\mathfrak h}\) a locally finite derivation. Then the Lie algebra \(\widehat{{\mathfrak h}}=\langle {\mathfrak h}, \partial_{\mathrm s}\rangle_{\mathop{\rm Lie}}\) also is solvable of derived length \(\le l+1\).
Proof. Since \({\mathfrak h}\) is invariant under \(\mathop{\rm ad}_{\partial}\), it also is invariant under \(\mathop{\rm ad}_{\partial_{\mathrm s}}\). Therefore, we have \[\widehat{{\mathfrak h}}^{(1)}=\langle {\mathop{\rm ad}}_{\partial_{\mathrm s}}({\mathfrak h}), \, {\mathfrak h}^{(1)} \rangle _{\mathop{\rm Lie}}\subset{\mathfrak h},\quad \widehat{{\mathfrak h}}^{(2)} \subset {\mathfrak h}^{(1)},\ldots, \widehat{{\mathfrak h}}^{(n)}\subset {\mathfrak h}^{(n+1)}\] for any natural number \(n\). Now the conclusion follows. ◻
The following simple observation will be useful in what follows.
Remark 8. If \({\mathfrak h}\) as in Question 1 contains a locally finite derivation \(\partial\) that is not locally nilpotent, then \(\widehat{{\mathfrak h}}=\langle {\mathfrak h}, \partial_{\mathrm s}\rangle_{\mathop{\rm Lie}}\) also verifies the assumptions of Question 1 and contains a semisimple derivation \(\partial_{\mathrm s}\neq 0\).
Corollary 9. The answer to Question 1 is affirmative if and only if it is affirmative under the additional assumption that \({\mathfrak h}\) is generated by a finite set consisting of locally nilpotent and semisimple derivations.
Proof. Let \({\mathfrak h}\subset\mathop{\rm Lie}(\mathop{\rm Aut}(X))\) be as in Question 1. According to the first reduction, \({\mathfrak h}=\langle \partial_1,\ldots,\partial_k\rangle_{\mathop{\rm Lie}}\), where the \(\partial_i\in{\mathfrak h}\) are locally finite. By Lemma 7, the Lie subalgebra \[\widehat{{\mathfrak h}}=\langle \partial_{1,\mathrm s}, \partial_{1,\mathrm n}, \ldots, \partial_{k,\mathrm s}, \partial_{k,\mathrm n} \rangle_{\mathop{\rm Lie}}\subset\mathop{\rm Lie}(\mathop{\rm Aut}(X))\] is solvable and generated by a finite set, consisting of semisimple and locally nilpotent derivations. Since \({\mathfrak h}\subset\widehat{{\mathfrak h}}\), the assertion follows. ◻
We use the following terminology, see for example [2].
Definition 10. A Lie subalgebra \({\mathfrak t}\subset\mathop{\rm Lie}(\mathop{\rm Aut}(X))\) is said to be toral if it consists of semisimple derivations. The rank \(\mathop{\rm rk}({\mathfrak h})\) is the maximal dimension of toral subalgebras of \({\mathfrak h}\).
We will need the following lemma.
Lemma 11 ([2]). For a toral Lie subalgebra \({\mathfrak t}\subset\mathop{\rm Der}({\mathcal{O}}(X))\), there exists a unique smallest torus \(T_{\min}=T_{\min}({\mathfrak t}) \subset \mathop{\rm Aut}(X)\) such that \({\mathfrak t}\subset \mathop{\rm Lie}(T_{\min})\). Every subspace \(E \subset\mathop{\rm Der}({\mathcal{O}}(X))\) invariant under \(\mathop{\rm ad}_{{\mathfrak t}}\) is also invariant under \(T_{\min}\). Furthermore, \({\mathfrak t}\) is locally finite and \(\dim({\mathfrak t})\le\dim(X)\). If \(\dim({\mathfrak t})=\dim(X)\), then \(X\) is a toric variety and \({\mathfrak t}\) is algebraic.
Definition 12 (see, e.g., [4]). A Lie subalgebra \({\mathfrak h}\subset\mathop{\rm Der}({\mathcal{O}}(X))\) is said to be algebraic (resp. integrable) if \({\mathfrak h}=\mathop{\rm Lie}(G)\) (resp. \({\mathfrak h}\subset \mathop{\rm Lie}(G)\)) for an algebraic subgroup \(G\subset\mathop{\rm Aut}(X)\).
A derivation \(\partial\in\mathop{\rm Der}({\mathcal{O}}(X))\) is called algebraic (resp. integrable) if the Lie subalgebra \({\Bbbk}\partial\) is algebraic (resp. integrable).
We know that \({\mathfrak h}\) is integrable if and only if it is locally finite, see [2]. In the latter case, \({\mathfrak h}\subset\mathop{\rm Lie}(\mathop{\rm Aut}(X))\). For example, every locally nilpotent derivation is algebraic, and every semisimple derivation is integrable. Indeed, if \(s\in\mathop{\rm Der}({\mathcal{O}}(X))\) is semisimple, then \(s\) is contained in a unique minimal algebraic torus \(T_{\min}(s)\subset\mathop{\rm Aut}(X)\), see [5].
Proposition 13 ([2]). Let \({\mathfrak h}\subset \mathop{\rm Lie}(\mathop{\rm Aut}(X))\) be the Lie subalgebra generated by a family of locally finite Lie subalgebras \({\mathfrak h}_i \subset\mathop{\rm Lie}(\mathop{\rm Aut}(X))\), \(i\in I\). Suppose that for all \(i\in I\), \({\mathfrak h}_i\) is algebraic, and let \({\mathfrak h}_i = \mathop{\rm Lie}(G_i)\) for a connected algebraic group \(G_i\subset \mathop{\rm Aut}(X)\). Then, \({\mathfrak h}\) is locally finite if and only if \({\mathfrak h}\) is finite-dimensional. In the latter case, the subgroup \(G\) generated by the \(G_i\) is algebraic and \({\mathfrak h}= \mathop{\rm Lie}(G)\).
Corollary 14. Consider a Lie subalgebra \[\label{eq:gener} {\mathfrak h}=\langle a_1,\ldots, a_k, b_1,\ldots,b_l\rangle_{\mathop{\rm Lie}} \subset \mathop{\rm Lie}(\mathop{\rm Aut}(X)),\qquad{(1)}\] where \(a_1,\ldots, a_k\) are locally nilpotent and \(b_1,\ldots,b_l\) are semisimple. Suppose that all generators \(b_i\) are algebraic, that is, for all \(i=k+1,\ldots, n\) we have \({\Bbbk}b_i=\mathop{\rm Lie}(T_i)\) for a one-dimensional algebraic torus \(T_i\subset\mathop{\rm Aut}(X)\). Then \({\mathfrak h}\) is locally finite if and only if it is finite-dimensional.
Proof. For \(i=1,\ldots,k\) we have \({\Bbbk}a_i=\mathop{\rm Lie}(U_i)\), where \(U_i=\exp(ta_i)\) is a one-parameter unipotent subgroup of \(\mathop{\rm Aut}(X)\). The assertion then follows from Proposition 13, which is itself a corollary of [2]. ◻
However, for \({\mathfrak h}\) in ?? , it is not known a priori whether the Lie subalgebra \[\langle {\Bbbk}a_1,\ldots,{\Bbbk}a_k, \mathop{\rm Lie}(T_{\min}(b_1)), \ldots, \mathop{\rm Lie}(T_{\min}(b_l))\rangle_{\mathop{\rm Lie}}\] is finite-dimensional (resp. solvable) if \({\mathfrak h}\) is finite-dimensional (resp. solvable).
Proposition 15. Let \({\mathfrak h}\subset \mathop{\rm Lie}(\mathop{\rm Aut}(X))\) be a solvable Lie subalgebra. Then \({\mathfrak h}\) is locally finite if and only if there exists a solvable connected algebraic subgroup \(G \subset \mathop{\rm Aut}(X)\) with derived length \(\le \dim(X) + 1\) such that \(h \subset \mathop{\rm Lie}(G)\). Such a minimal algebraic subgroup \(G_{\min}({\mathfrak h})\) is unique.
Recall that the derived ideal \({\mathfrak g}^{(1)}=[{\mathfrak g},{\mathfrak g}]\) of a solvable finite-dimensional Lie algebra \({\mathfrak g}\) is nilpotent, see [6]. This is no longer true, in general, if \({\mathfrak g}\) is infinite-dimentional, see, e.g., Example 18 below. We have the following criterion for the finite-dimensionality of a solvable Lie algebra.
::: {#Lemma 1 .lem} Lemma 16. Let \({\mathfrak h}\) be a finitely generated solvable Lie algebra. Then \({\mathfrak h}\) is finite-dimensional if and only if the derived ideal \({\mathfrak h}^{(1)}\) is nilpotent and finitely generated. :::
Proof. The quotient \({\mathfrak h}/{\mathfrak h}^{(1)}\) is a finitely generated abelian Lie algebra. Therefore, it is finite-dimensional. Thus, \({\mathfrak h}\) is finite-dimensional if and only if \({\mathfrak h}^{(1)}\) is. If \({\mathfrak h}\) is finite-dimensional, then \({\mathfrak h}^{(1)}\) is nilpotent and finitely generated.
In the opposite direction, suppose that \({\mathfrak h}^{(1)}\) is \(k\)-step nilpotent and finitely generated. Let \(e_1,\ldots,e_n\) be a set of generators of \({\mathfrak h}^{(1)}\). Then \[{\mathfrak h}^{(1)}=\mathop{\rm span}\big([e_{i_1},[e_{i_2}, \ldots,[e_{i_{k-1}},e_{i_k}]]\ldots]]\,|\,i_1, \ldots,i_k\in\{1,\ldots,n\}\big)\] is finite-dimensional. Therefore, also \({\mathfrak h}\) is finite-dimensional. ◻
By the first reduction, Proposition 3 admits the following equivalent formulation.
::: {#Prop 3 .prop} Proposition 17. Let \({\mathfrak h}=\langle a_1,\ldots,a_k\rangle_{\mathop{\rm Lie}}\) be a solvable Lie subalgebra of \(\mathop{\rm Lie}(\mathop{\rm Aut}(X))\) generated by the locally finite derivations \(a_i\). Then \({\mathfrak h}\) is finite dimensional if and only if the derived ideal \({\mathfrak h}^{(1)}\) is nilpotent. :::
Proof. In the sense of “only if”, our assertion is classical. Conversely, suppose that \({\mathfrak h}^{(1)}\) is nilpotent. According to Lemma 16, it suffices to show that \({\mathfrak h}^{(1)}\) is finitely generated. By setting \({\mathfrak h}_i=\langle a_1,\ldots,a_i\rangle_{\mathop{\rm Lie}}\), we proceed by induction on \(i\). Since \({\mathfrak h}_1={\Bbbk}a_1\) is abelian, we have \({\mathfrak h}_1^{(1)}=0\). Suppose that \({\mathfrak h}_i^{(1)}\) is finitely generated for some \(i\in\{1,\ldots,k-1\}\). By Lemma 16, then \({\mathfrak h}_i\) is finite-dimensional. We have \[{\mathfrak h}_{i+1}^{(1)}=\langle {\mathfrak h}_i^{(1)}, \,{\mathop{\rm ad}}_{a_{i+1}}({\mathfrak h}_i)\rangle_{\mathop{\rm Lie}}.\] Recall that \(\mathop{\rm ad}_{a_{i+1}}\in \mathop{\rm End}({\mathfrak h})\) is locally finite, see, e.g., [2]. It follows that \({\mathfrak h}_i^{(1)}\subset{\mathfrak h}_i\) and \({\mathop{\rm ad}}_{a_{i+1}}({\mathfrak h}_i)\) are finite-dimensional. Therefore, \({\mathfrak h}_{i+1}^{(1)}\) is finitely generated. This gives the induction step. ◻
Example 18 ([1]). Let \({\mathfrak h}=\langle a_0, \,a_1,\ldots, a_i,\ldots\rangle_{\mathop{\rm Lie}}\), where \([a_0,a_i]=a_{i+1}\) for all \(i\ge 1\) and \([a_i,a_j]=0\) for \(i,j\ge 1\). Then \({\mathfrak h}^{(1)}=\oplus_{i=1}^{\infty} {\Bbbk}a_i\) is abelian and infinite dimensional. Here, \(\mathop{\rm ad}_{a_0}\) is not locally finite. We can realize \({\mathfrak h}\) as a Lie algebra of derivations by setting \[X={\mathbb{A}}^1\times ({\mathbb{A}}^1\setminus\{0\}) =\mathop{\rm Spec}{\Bbbk}[x,y,y^{-1}],\quad a_0=\partial/\partial y, \quad\text{and}\quad a_i=y^{-i}\partial/\partial x, \quad i=1,2,\ldots.\]
Due to Remark 8, the following result implies Theorem 4:
Theorem 19. Let \({\mathfrak h} \subset\mathop{\rm Lie}(\mathop{\rm Aut}({\mathbb{A}}^2))\) be a solvable Lie subalgebra generated by locally finite derivations \(\partial_1,\ldots,\partial_n\). Suppose that \({\mathfrak h}\) contains a semisimple derivation \(\delta\neq 0\). Then \({\mathfrak h}\) is locally finite and triangulable2.
See subsection 3.5 for the proof.
We use the following notation.
Notation 20. We define \[{\mathfrak u}^+_2=\{\partial\in\mathop{\rm Der}({\Bbbk}[x,y])\,|\,\partial= p^+(y)\partial/\partial x+c^+\partial/\partial y\}\] resp. \[{\mathfrak u}_2^-=\{\partial\in\mathop{\rm Der}({\Bbbk}[x,y])\,|\,\partial= p^-(x)\partial/\partial y+c^-\partial/\partial x\},\] where \(p^\pm\in{\Bbbk}[t]\) and \(c^\pm\in{\Bbbk}\). It is well known that \({\mathfrak u}_2^+\) (resp. \({\mathfrak u}^-_2\)) is a metabelian (i.e., 2-step solvable) subalgebra of \(\mathop{\rm Lie}(\mathop{\rm Aut}({\mathbb{A}}^2))\) of infinite dimension. It consists of locally nilpotent upper (resp. lower) triangular derivations. The Lie algebra \({\mathfrak u}_2^\pm\) is filtered by a sequence of locally finite Lie subalgebras \[{\mathfrak u}_{2,\,\le d}^\pm=\{\partial\in{\mathfrak u}^\pm_2\,|\, \mathop{\rm deg}(p^\pm)\le d\}.\] The group \(U_2^\pm=\exp({\mathfrak u}_2^\pm)\) is an infinite-dimensional nested unipotent subgroup of \(\mathop{\rm Aut}({\mathbb{A}}^2)\) filtered by the unipotent algebraic subgroups \((U_2^\pm)_{\le d}=\exp({\mathfrak u}_{2,\,\le d}^\pm)\).
Consider also a maximal toral Lie subalgebra \[{\mathfrak t}_2={\Bbbk}x\partial/\partial x\oplus{\Bbbk}y\partial/\partial y\subset \mathop{\rm Der}({\Bbbk}[x,y]).\] The semidirect product \({\mathfrak j}_2^\pm={\mathfrak u}^\pm_2\rtimes {\mathfrak t}_2\) is the Lie algebra of the de Jonquières group \(\mathop{\rm JONQ}^\pm({\mathbb{A}}^2)\) of upper (lower) triangular automorphisms of \({\mathbb{A}}^2\), respectively. Note that \({\mathfrak j}_2^+\) and \({\mathfrak j}_2^-\) are conjugate by \(\mathop{\rm Ad}_{\tau}\), where \(\tau\colon (x,y)\mapsto (y,x)\).
Definition 21. A Lie subalgebra \({\mathfrak h}\subset\mathop{\rm Der}({\Bbbk}[x,y])\) is said to be upper (resp. lower) triangular, if \({\mathfrak h}\subset{\mathfrak j}_2^+\) (resp. \({\mathfrak h}\subset{\mathfrak j}_2^-\)). It is said to be triangulable, if \({\mathfrak h}\) is \(\mathop{\rm Ad}\)-conjugate to a triangular Lie subalgebra.
Lemma 22. Let \({\mathfrak h}\subset\mathop{\rm Lie}(\mathop{\rm Aut}({\mathbb{A}}^2))\) be a locally finite solvable subalgebra. Then \({\mathfrak h}\) is triangulable and its derived length is at most \(3\). The derived ideal \([{\mathfrak h},{\mathfrak h}]\) is either abelian or two-step nilpotent.
Proof. Since \({\mathfrak h}\) is locally finite and solvable, it is integrable and is contained in a locally integrable Borel subalgebra \({\mathfrak b}\) of \(\mathop{\rm Lie}(\mathop{\rm Aut}({\mathbb{A}}^2))\), see [1]. By [1], \({\mathfrak b}=\mathop{\rm Lie}(B)\) for some Borel subgroup \(B\) of \(\mathop{\rm Aut}({\mathbb{A}}^2)\). It is well known that any Borel subgroup of \(\mathop{\rm Aut}({\mathbb{A}}^2)\) is conjugate to the triangular subgroup \(\mathop{\rm JONQ}^+({\mathbb{A}}^2)\), see, e.g., [7] and the references therein. Therefore, \({\mathfrak b}\) is \(\mathop{\rm Ad}\)-conjugate to the Borel subalgebra of triangular derivations \[{\mathfrak j}_2^+= \mathop{\rm Lie}({\mathop{\rm JONQ}}^+({\mathbb{A}}^2))= {\Bbbk}[y]\partial/ \partial x\oplus {\Bbbk}x\partial/ \partial x\oplus {\Bbbk}y\partial/ \partial y \oplus{\Bbbk}\partial/ \partial y,\] see [1]. Note that \({\mathfrak j}_2^+\) is of derived length 3; indeed, the derived ideal \[[{\mathfrak j}_2^+,{\mathfrak j}_2^+]={\mathfrak u}_2^+ ={\Bbbk}[y]\partial/ \partial x\oplus {\Bbbk}\partial/ \partial y\] is two-step nilpotent. ◻
Remarks 23. \(\,\)
1. A Lie subalgebra \({\mathfrak h}\subset\mathop{\rm Lie}(\mathop{\rm Aut}({\mathbb{A}}^2))\) is said to be locally integrable if it is filtered by locally finite Lie subalgebras. A solvable locally integrable Lie subalgebra is triangulable, see [1] and the subsequent discussion.
2. In higher dimensions, an analogue of Lemma 22 is generally not valid. For example, by Bass [8], the \({\mathbb{G}_{\mathrm a}}\)-subgroup \(U=\exp({\Bbbk}\partial)\subset\mathop{\rm Aut}({\mathbb{A}}^3)\), where \(\partial\in\mathop{\rm Der}({\Bbbk}[x,y,z])\) is the locally nilpotent Nagata derivation, is not triangulable. Consequently, the abelian Lie subalgebra \({\Bbbk}\partial\) is also not triangulable. In fact, it is not even stably triangulable, see [2]. This provides similar examples in all dimensions \(n\ge 3\).
In what follows, we use the following notation, cf., e.g., [2] and [1].
Notation 24. Let \[\Lambda=\{(a,b)\in{\mathbb{Z}}^2\,| \,a,b\ge -1,\,\,\, (a,b)\neq (-1,-1)\}.\] For \((a,b)\in\Lambda\), we define: \[\partial_{a,b}= (b+1)x^{a+1}y^b\partial/\partial x -(a+1)x^ay^{b+1}\partial/\partial y\] and \[{\rm bideg}(\partial_{a,b})=(a,b).\] This defines a bigrading on the Lie subalgebra \({\rm Vec}^0({\mathbb{A}}^2)\subset \mathop{\rm Lie}(\mathop{\rm Aut}({\mathbb{A}}^2))\) of zero-divergence derivations. Indeed, we have the following commutation relations: \[\label{eq:comm} [\partial_{a,b},\partial_{a',b'}]= \det\left(\begin{matrix} a'+1&a+1\\b'+1&b+1\end{matrix}\right) \partial_{a+a', b+b'},\tag{1}\] where \[(a,b),\,(a',b')\in\Lambda \,\,\,\text{and} \,\,\, \partial_{-1,-1}:=0.\] All graded pieces of \({\rm Vec}^0({\mathbb{A}}^2)\) are one-dimensional, and therefore \[{\rm Vec}^0({\mathbb{A}}^2)= \bigoplus_{(a,b)\in\Lambda} {\Bbbk}\partial_{a,b}, \quad\text{where}\quad [ {\Bbbk}\partial_{a,b}, {\Bbbk}\partial_{c,d}]\subset {\Bbbk}\partial_{a+c,b+d}.\] This bigrading respects the natural bigrading of \({\Bbbk}[x,y]\), namely: \[\partial_{a,b}(x^iy^j)=\det\left(\begin{matrix} i&a+1\\j&b+1\end{matrix}\right) x^{i+a}y^{j+b}.\]
Consider the Lie subalgebra of the derivations with constant divergence \[\mathop{\rm Lie}(\mathop{\rm Aut}({\mathbb{A}}^2))= {\rm Vec}^{\mathrm c}({\mathbb{A}}^2) ={\rm Vec}^0({\mathbb{A}}^2) \oplus {\Bbbk}\partial_{\rm Eul}, \quad\text{where}\quad \partial_{\rm Eul}=x\partial/\partial x+y\partial/\partial y\] is the Euler derivation. Indeed, every \(\partial\in{\rm Vec}^{\mathrm c}({\mathbb{A}}^2)\) admits a unique decomposition \[\partial=c_0\partial_{\rm Eul}+ {\sum}_{i,j}c_{i,j} \partial_{i,j},\] where the sum is finite.
The Lie algebra \({\rm Vec}^{\mathrm c}({\mathbb{A}}^2)\) includes the maximal two-dimensional toral subalgebra \[{\mathfrak t}_2=\{\delta_{\alpha,\beta}\,| \,(\alpha,\beta)\in{\Bbbk}^2\}, \quad\text{where}\quad \delta_{\alpha,\beta}=\alpha x\partial/\partial x +\beta y\partial/\partial y.\] The Lie algebra \({\rm Vec}^{\mathrm c}({\mathbb{A}}^2)\) is bigraded with one-dimensional graded pieces, except for the unique two-dimensional graded piece \({\mathfrak t}_2\) of weight \((0,0)\). The additional commutation relations are as follows: \[\label{eq:comrel} [\delta_{\alpha, \beta}, \delta_{\gamma, \eta}] = 0\quad\text{and}\quad [\delta_{\alpha, \beta}, \partial_{a,b}]= (\alpha a + \beta b) \partial_{a, b}.\tag{2}\] Thus, every derivation \(\partial_{a, b}\) is an eigenvector of \(\mathop{\rm ad}_{\delta_{\alpha, \beta}}\in \mathop{\rm End}({\rm Vec}^0({\mathbb{A}}^2))\).
Definitions 25. \(\,\)
1. We call the lattice points \((-1,n)\) and \((m,-1)\) with \(m,n\ge 0\) Demazure points. They correspond to the homogeneous locally nilpotent derivations \[\partial_{-1,n}=(n+1)y^n\partial/ \partial x \quad\text{resp.}\quad \partial_{m,-1}=-(m+1)x^m\partial/ \partial y.\]
2. Given a derivation \(\partial_0=\sum_{i,j} c_{i,j}\partial_{i,j} \in\mathop{\rm Vec}^0({\mathbb{A}}^2)\), the Newton polygon \(N(\partial_0)\) is the convex hull of \[\mathop{\rm supp}(\partial_0):=\{(i,j) \in{\mathbb{Z}}^2\,|\,c_{i,j}\neq 0\}.\] If \(\partial=\partial_0+c_0\partial_{\rm Eul} \in\mathop{\rm Vec}^{\mathrm c}({\mathbb{A}}^2)\), where \(c_0\neq 0\), then we let \(N(\partial)\) be the convex hull of \(\mathop{\rm supp}(\partial_0)\cup (0,0)\).
Using the bigrading on \(\mathop{\rm Der}({\Bbbk}[x,y])\), one can easily deduce the following corollary, cf. [9], [10], and [11].
Corollary 26. For every locally finite derivation \(\partial\in(\mathop{\rm Der}({\Bbbk}[x,y])\), all vertices of the Newton polygon \(N(\partial)\) are among the lattice points \((-1,j), (i,-1)\) and \((0,0)\), where \(i,j\ge 0\). If \(\partial\) is locally nilpotent, then all vertices of \(N(\partial)\) are Demazure points.
We begin with the following observations.
Remark 27. Let \(s\in{\mathfrak h}\) be a semisimple derivation. The toral subalgebra \({\Bbbk}s\subset\mathop{\rm Lie}(\mathop{\rm Aut}({\mathbb{A}}^2))\) is contained in \(\mathop{\rm Lie}(T_{\min}(s))\) for a minimal algebraic torus \(T_{\min}(s)\subset\mathop{\rm Aut}({\mathbb{A}}^2)\), see Lemma 11. If \(s\) is not algebraic, we have \(\dim(T_{\min}(s))=2\). In this case, according to the Białynicki-Birula theorem [12], \(T_{\min}(s)\) is conjugate in \(\mathop{\rm Aut}({\mathbb{A}}^2)\) to the standard diagonal 2-torus \({\mathbb{T}}\), and \({\Bbbk}s\) is conjugate to a Lie subalgebra \({\Bbbk}\delta_{\alpha,\beta}\), where \(\alpha,\beta\in{\Bbbk}\) are nonzero with \(\alpha/\beta\in{\Bbbk}\setminus{\mathbb{Q}}\).
In the case where \(T_{\min}(s)\) is a one-dimensional algebraic torus, that is, \(s\) is algebraic, \(T_{\min}(s)\) is conjugate to a subtorus of the diagonal \(2\)-torus \({\mathbb{T}}\), see [12] or [13]. Therefore, \({\Bbbk}s\) is \(\mathop{\rm Ad}\)-conjugate to \({\Bbbk}\delta_{m,n}\) for a pair \((m,n)\) of coprime integers,
Given a pair \((\alpha,\beta)\in {\mathbb{A}}^2\setminus \{(0,0)\}\), the action of the semisimple derivation \(\mathop{\rm ad}_{\delta_{\alpha,\beta}}\) on \({\mathop{\rm Vec}}^{\mathrm c}({\mathbb{A}}^2)\) leads to a spectral decomposition \[{\mathop{\rm Vec}}^{\mathrm c}({\mathbb{A}}^2)= \bigoplus_{\lambda\in{\Bbbk}} E_\lambda(\delta_{\alpha, \beta}),\] where \(E_\lambda(\delta_{\alpha, \beta})\) is the eigenspace of \(\mathop{\rm ad}_{\delta_{\alpha, \beta}}\) associated with the eigenvalue \(\lambda\) (see 2 ). The eigenspace \(E_0(\delta_{\alpha, \beta})\) coincides with the centralizer of \(\delta_{\alpha, \beta}\) in \({\mathop{\rm Vec}}^{\mathrm c}({\mathbb{A}}^2)\). If \(\delta_{\alpha,\beta}\in{\mathfrak h}\), where \({\mathfrak h}\) is a Lie subalgebra of \({\mathop{\rm Vec}}^{\mathrm c}({\mathbb{A}}^2)\), then we have \[\label{eq:dir-dec} {\mathfrak h}=\bigoplus_{\lambda\in{\Bbbk}}\left({\mathfrak h}\cap E_\lambda(\delta_{\alpha,\beta})\right).\tag{3}\]
We need the following simple lemma.
Lemma 28 ([1]). Let \(V\) be a finite-dimensional vector space over \({\Bbbk}\) and \(A\) a diagonalizable endomorphism of \(V\). Consider the decomposition \(V = \bigoplus_\lambda V_\lambda\), where the \(V_\lambda\) are the eigenspaces of \(A\) associated with the different eigenvalues \(\lambda\in{\Bbbk}\). Let a subspace \(U \subset V\) be invariant under \(A\), and let \(u=\sum_\lambda u_\lambda \in U\), where \(u_\lambda \in V_\lambda\). Then \(u_\lambda \in U\) for all \(\lambda\).
Lemma 29. Given \(h\in{\mathfrak h}\), consider the spectral decomposition \(h=\sum_\lambda h_\lambda\) associated with \(\delta_{\alpha,\beta}\in{\mathfrak h}\), where \(h_\lambda\in {\mathfrak h}\cap E_\lambda(\delta_{\alpha,\beta})\), see 3 . Then the set of nonzero components \(h_\lambda\) is finite, and \(h_\lambda\in{\mathfrak h}\) for all \(\lambda\).
Proof. The decomposition \[h=\delta_{\eta,\zeta}+ \sum_{k,l} c_{k,l}\partial_{k,l}\] contains only a finite number of nonzero homogeneous components. By 2 , every such component is an eigenvector of \(\mathop{\rm ad}_{\delta_{\alpha,\beta}}\). Therefore, \(h\) belongs to a finite-dimensional vector subspace \[U={\Bbbk}\delta_{\eta,\zeta} \oplus\bigoplus_{k,l} {\Bbbk}\partial_{k,l}\subset{\mathfrak h}\] invariant under \(\mathop{\rm ad}_{\delta_{\alpha,\beta}}\). The assertions then follow from Lemma 28. ◻
Lemma 30. Let \({\mathfrak h}\subset \mathop{\rm Lie}(\mathop{\rm Aut}({\mathbb{A}}^2))\) be a Lie subalgebra. Suppose that \(\delta_{\alpha,\beta}\in{\mathfrak h}\) for some \(\alpha,\beta\neq 0\) such that \(\alpha/\beta\notin{\mathbb{Q}}\), that is, \(\delta_{\alpha,\beta}\) is not algebraic. Let \[\partial=c_{0,0}\delta_{\eta,\zeta}+ \sum_{(i,j)\in\mathop{\rm supp}(\partial)\setminus (0,0)} c_{i,j}\partial_{i,j}\in{\mathfrak h}.\] Then \(\partial_{i,j}\in{\mathfrak h}\) for all \((i,j)\in\mathop{\rm supp}(\partial)\setminus (0,0)\). In particular, \(\partial_{i,j}\in{\mathfrak h}\) for any vertex \(v\neq (0,0)\) of Newton’s polygon \(N(\partial)\).
Proof. Recall that every \(\partial_{i,j}\) is an eigenvector of \(\delta_{\alpha,\beta}\) associated with the eigenvalue \(\lambda=\alpha i+ \beta j\), see 2 . Since \(\alpha/\beta\notin{\mathbb{Q}}\), for \(\lambda \neq 0\) the corresponding eigenspace \(E_\lambda\) of \(\mathop{\rm ad}_{\delta_{\alpha,\beta}}\in \mathop{\rm End}(\mathop{\rm Der}{\Bbbk}[x,y])\) is one-dimensional: \(E_\lambda={\Bbbk}\partial_{i,j}\). Since the subspace \({\mathfrak h}\subset \mathop{\rm Der}{\Bbbk}[x,y]\) is invariant under \(\mathop{\rm ad}(\delta_{\alpha,\beta})\), the lemma follows from Lemma 28. ◻
Let \((\alpha,\beta)=(-m,n)\), where \(m,n\ge -1\) are relatively prime integers such that \((n,m)\notin \{(0,0), (-1,-1)\}\). Let’s determine the eigenspaces \(E_\lambda(\mathop{\rm ad}_{\delta_{-m, n}})\). By 2 , the eigenvalues of \(\mathop{\rm ad}_{\delta_{-m, n}}\) are integers. Therefore, \[\label{eq:decomp} {\mathop{\rm Vec}}^{\mathrm c}({\mathbb{A}}^2)= \bigoplus_{\lambda\in{\mathbb{Z}}} E_\lambda(\delta_{-m, n}).\tag{4}\] For \(\lambda\in{\mathbb{Z}}\setminus\{0\}\) the corresponding eigenspace is \[E_\lambda(\delta_{-m, n})= \bigoplus_{nl-mk=\lambda}{\Bbbk}\partial_{k,l}= \bigoplus_{i}{\Bbbk}\partial_{k_0+in,l_0+im},\] where the integers \(k_0,l_0\) and \(i\) satisfy \[\label{eq:rel} nl_0-mk_0 =\lambda,\quad k_0+i n\ge -1,\quad l_0+i m\ge-1, \quad (k_0+i n, l_0+i m) \notin\{(-1,-1),(0,0)\}.\tag{5}\]
The centralizer \(E_0(\delta_{-m, n})\) of \(\delta_{-m, n}\) in \({\mathop{\rm Vec}}^{\mathrm c}({\mathbb{A}}^2)\) is described in the following lemma. The proof of this lemma is a simple calculation, and we therefore omit it.
Lemma 31. For \(n_0,m_0\ge 2\) we have \[\label{eq:eigen-1} E_0(\delta_{m_0, 1}) =E_0(\delta_{-m_0, -1}) ={\mathfrak t}_2\oplus {\Bbbk}\partial_{-1,m_0}\qquad{(2)}\] and \[\label{eq:eigen-2} E_0(\delta_{1, n_0})= E_0(\delta_{-1, -n_0}) ={\mathfrak t}_2\oplus {\Bbbk}\partial_{n_0,-1}.\qquad{(3)}\] For a pair of coprime integers \((m,n)\notin\{\pm (m_0, 1),\, \pm (1, n_0)\,|\,m_0,n_0\ge 2\}\) we have \[\label{eq:eigen} E_0(\delta_{-m, n})=E_0(\delta_{m, -n})= {\mathfrak t}_2\oplus \bigoplus_{nl-mk=0}{\Bbbk}\partial_{k,l}= {\mathfrak t}_2\oplus\bigoplus_{i=1}^{\infty} {\Bbbk}\partial_{in,im}.\qquad{(4)}\]
Let’s introduce the following notion.
Definition 32. A pair of derivations \((s,\partial_{\mathrm n})\), where \(s\) is semisimple and \(\partial_{\mathrm n}\) is locally nilpotent, is said to be opportune if it verifies the relations \[[\partial_{\mathrm n}, s]\neq 0 \quad\text{and}\quad [\partial_{\mathrm n}, [\partial_{\mathrm n}, s]]=0.\] Note then if \(\mathop{\rm ad}_s(\partial_{\mathrm n})= \lambda\partial_{\mathrm n}\), where \(\lambda\neq 0\), then the pair \((s,\partial_{\mathrm n})\) is opportune.
Lemma 33. Suppose that a Lie subalgebra \({\mathfrak h}\subset\mathop{\rm Lie}(\mathop{\rm Aut}({\mathbb{A}}^2))\) contains an opportune pair \((s,\partial_{\mathrm n})\). Then \(\mathop{\rm rk}({\mathfrak h})=2\).
Proof. Consider the one-parameter unipotent subgroup \(U=\exp({\Bbbk}\partial_{\mathrm n}) \subset\mathop{\rm Aut}({\mathbb{A}}^2)\). The orbit \({\mathop{\rm Ad}}_U(s)\subset{\mathfrak h}\) in the adjoint representation of \(U\) consists of semisimple elements \(s_t\), where \[s_t = {\mathop{\rm Ad}}_{\exp(t\partial_{\mathrm n})}(s) = s + t [\partial_{\mathrm n},s],\quad t\in{\Bbbk},\] see [10]. We have \([s_t, s_{t'}]=0\) and \(s_t \neq s\) for \(t\neq 0\).
Assuming that \([\partial_{\mathrm n},s]= \lambda s\) for some \(\lambda\neq 0\), we arrive at a contradiction with our assumption that \(\partial_{\mathrm n}\) is locally nilpotent. Therefore, \(s\) and \([\partial_{\mathrm n},s]\) are linearly independent. Then, \(s=s_0\) and \(s_1\) are also linearly independent. Thus, \({\mathfrak h}\) contains a two-dimensional toral subalgebra \[{\mathfrak t}=\langle s_0, s_1\rangle_{\mathop{\rm Lie}} = {\Bbbk}s_0\oplus {\Bbbk}s_1.\] Indeed, every nonzero \(\delta\in {\mathfrak t}\) is semisimple, being proportional to some \(s_t\). ◻
To detect an opportune pair \((s,\partial_{\mathrm n})\) (not necessarily contained in our Lie algebra \({\mathfrak h}\)), we use the following result.
Lemma 34. Let a Lie subalgebra \({\mathfrak h}\subset\mathop{\rm Lie}(\mathop{\rm Aut}(X))\) contains a pair \((\delta, \partial)\), where \(\delta\) is semisimple and \(\partial\) is locally finite. If \([\delta, \partial]\neq 0\), then there exists \(\lambda\in \mathop{\rm Spec}(\delta)\setminus\{0\}\) such that \(\partial_{\lambda}\) is locally nilpotent and \([\delta, \partial_{\lambda}]\neq 0\), while \([\partial_{\lambda},[\partial_{\lambda},\delta]]=0\). So, \((\delta,\partial_{\lambda})\) is an opportune pair.
Proof. Up to a suitable coordinate change on \({\mathbb{A}}^2\), we can assume that \(\delta=\delta_{\alpha,\beta}\), see Remark 27. Let us consider the spectral decomposition of \(\partial\) according to 4 : \[\partial=\delta_{\xi,\eta}+ \sum_{i,j} c_{i,j}\partial_{i,j} =\sum_{\lambda\in\sigma} \partial_{\lambda},\] where \(\sigma\subset\mathop{\rm Spec}(\delta)\) is finite, is not reduced to the singleton \(\{0\}\), and \(\partial_{\lambda}\in E_{\lambda}(\delta)\setminus\{0\}\) for all \(\lambda\in\sigma\). So, for \(\lambda\in \sigma\setminus\{0\}\) we have \(\partial_{\lambda}=\sum_{\alpha i+\beta j =\lambda} c_{i,j}\partial_{i,j}\).
Let \({\mathbb{Q}}\subset K\) be the finite extension spanned by \(\sigma\) over \({\mathbb{Q}}\) in \({\Bbbk}\). We can embed \(K\) in \({\mathbb{C}}\) as a subfield. Let \({\mathcal{N}}(\partial)\) be the convex hull of \(\sigma\) in \({\mathbb{C}}\). The vertices of this polygon belong to \(\sigma\).
Consider the linear function \(f(x,y)=\alpha x+\beta y\) on \({\mathbb{R}}^2\) with values in \({\Bbbk}\). For any \((i,j)\in\mathop{\rm supp}(\partial)\), we have \(\lambda=f(i,j)\in\sigma\). Therefore, \(f\colon K\to{\mathbb{C}}\) sends the Newton polygon \(N(\partial)\) onto \({\mathcal{N}}(\partial)\). Since \([\delta, \partial]\neq 0\), \(N(\partial)\) is not contained in the line \(\{f=0\}\), in particular \({\mathcal{N}}(\partial)\neq\{0\}\).
Choose a vertex \(v=(i,j)\) of \(N(\partial)\) such that \(f(v)\neq 0\) and \(\lambda=f(v)\in\sigma\) is a vertex of \({\mathcal{N}}(\partial)\), and let \({\mathcal{L}}\colon{\mathbb{C}}\to{\mathbb{R}}\) be a linear function that attains its maximum on \({\mathcal{N}}(\partial)\) at \(\lambda\). Then \({\mathcal{L}}\circ f\colon{\mathbb{Z}}^2\to{\mathbb{R}}\) defines a grading on the Lie algebra \(\mathop{\rm Vec}^{\mathrm c}({\mathbb{A}}^2)\). With respect to this grading, \(\partial_{\lambda}\) represents the principal part of \(\partial\). Since \(\partial\) is locally finite, \(\partial_{\lambda}\) is locally nilpotent and possesses the desired properties, see, e.g., [9], or [10], or [11]. ◻
The following corollary is immediate.
Corollary 35. Let \({\mathfrak h}\subset\mathop{\rm Lie}(\mathop{\rm Aut}(X))\) be a Lie subalgebra generated by locally finite derivations \(\partial_1,\partial_2,\ldots\). If \({\mathfrak h}\) contains a semisimple derivation \(\delta\neq 0\), then either the Lie subalgebra \({\Bbbk}\delta\) is central in \({\mathfrak h}\), or there exists \(\partial\in{\mathfrak h}\) and a nonzero eigenvalue \(\lambda\in \mathop{\rm Spec}(\mathop{\rm ad}_{\delta})\) such that \((\delta, \partial_{i,\lambda})\) is an opportune pair.
Proof. If \([\delta, \partial_i]=0\) for all \(i\), then \({\Bbbk}\delta\) is central in \({\mathfrak h}\). Otherwise, \([\delta, \partial_i]\neq 0\) for some index \(i\), and therefore there exists \(\lambda\in \mathop{\rm Spec}(\delta)\setminus\{0\}\) such that \(\partial_{i,\lambda}\in{\mathfrak h}\) is locally nilpotent and \((\delta, \partial_{i,\lambda})\) is an opportune pair, see Lemma 34. ◻
Corollary 36. Let \({\mathfrak h}\subset\mathop{\rm Lie}(\mathop{\rm Aut}({\mathbb{A}}^2))\) be a solvable Lie subalgebra generated by locally finite derivations. Suppose that \({\mathfrak h}\) is not toral. Then \(\mathop{\rm rk}({\mathfrak h})=2\) if and only if \({\mathfrak h}\) contains an opportune pair.
Proof. If \({\mathfrak h}\) contains an opportune pair, then \(\mathop{\rm rk}({\mathfrak h})=2\) by Lemma 33. Conversely, suppose that \(\mathop{\rm rk}({\mathfrak h})=2\), and let us show that \({\mathfrak h}\) contains an opportune pair.
By choosing appropriate coordinates in \({\mathbb{A}}^2\), we can assume that \({\mathfrak t}_2\subset{\mathfrak h}\), see Remark 27. Let \(\delta_{\alpha,\beta}\in{\mathfrak t}_2\) be a non-algebraic semisimple derivation. Since \(\alpha/\beta\) is non-rational, every \(\partial_{a,b}\) with \((a,b)\neq (0,0)\) is an eigenvector of \(\delta\) associated with a nonzero eigenvalue, see 2 . There exists a locally finite generator \(\partial_i\notin{\mathfrak t}_2\) of \({\mathfrak h}\). All homogeneous components \(c_{a,b}\partial_{a,b}\) of \(\partial_i\) belong to \({\mathfrak h}\), see Lemma 30. Since \(\partial_i\) is locally finite and \(\partial_i\notin{\mathfrak t}_2\), it has a locally nilpotent homogeneous component \(\partial=c_{a,b}\partial_{a,b}\in{\mathfrak h}\), where \(c_{a,b}\neq 0\) and \(\min\{a,b\}=-1\), see Corollary 26. By 2 , \([\delta_{\alpha,\beta},\partial]\neq 0\). Therefore, \((\delta_{\alpha,\beta},\partial)\) is an opportune pair from \({\mathfrak h}\), see Lemma 34. ◻
In what follows we use the following lemma.
Lemma 37. A solvable Lie subalgebra \({\mathfrak h}\subset \mathop{\rm Lie}(\mathop{\rm Aut}({\mathbb{A}}^2))\) cannot contain a pair of locally nilpotent homogeneous derivations \(\partial_{-1,k}\) and \(\partial_{l,-1}\) with \(k,l\ge 1\).
Proof. We reason by contradiction. First, suppose that \({\mathfrak h}^{(1)}\) contains \(\partial_{-1,k}\) and \(\partial_{l,-1}\) for some \(k\ge 2\) and \(l\ge 1\). Using the commutation relations 1 , we deduce by induction that \[{\mathop{\rm ad}}^n_{\partial_{l,-1}}(\partial_{-1,k})= c_n\partial_{nl-1,k-n}\in{\mathfrak h}^{(1)}, \quad\text{where} \quad c_n\neq 0\quad \text{for}\quad n=1,\ldots,k+1.\] In particular, \(\partial_{l_1,-1}\in{\mathfrak h}^{(1)}\), where \(l_1:=(k+1)l-1>l\). Similarly, we obtain: \(\partial_{-1,k_1}\in {\mathfrak h}^{(1)}\), where \(k_1:=(l_1+1)k-1>k\). Continuing in this way, we conclude that \({\mathfrak h}^{(1)}\) contains infinite sequences of locally nilpotent derivations \(\partial_{-1,k_i}\) and \(\partial_{l_i,-1}\) with strictly increasing indices \(k_i\) and \(l_i\). Then \({\mathfrak h}^{(2)}\) contains an infinite set of homogeneous derivations \(\partial_{l_j-1,k_i-1} =c[\partial_{-1,k_i},\partial_{l_i,-1}]\), where \(c\neq 0\), \({\mathfrak h}^{(3)}\) contains an infinite set of homogeneous derivations \(\partial_{l_i+l_j-2,k_i+k_j-2}\), etc. Finally, \({\mathfrak h}^{(n)}\neq 0\) for all \(n\), contrary to our assumption that \({\mathfrak h}\) is solvable.
In the case where \(k=l=1\), \({\mathfrak h}\) contains \(\partial_{-1,1}= \frac{1}{2}y\partial/\partial x\) and \(\partial_{1,-1}=-\frac{1}{2}x\partial/\partial y\). Therefore, \({\mathfrak h}\) contains the \({\mathfrak s}{\mathfrak l}_2\)-subalgebra \(\langle y\partial/\partial x, x\partial/\partial y\rangle_{\mathop{\rm Lie}}\), contrary to our assumption. ◻
The next example illustrates Lemma 37.
Example 38. Consider the locally nilpotent derivation \(\partial\) of \({\Bbbk}[x,y]\), where \[\partial=(x-y)^2(\partial/\partial x+\partial/\partial y).\] Also consider the Lie subalgebra \[{\mathfrak h}= \langle \partial, x\partial/\partial x, y\partial/\partial y\rangle_{\mathop{\rm Lie}} =\langle \partial, {\mathfrak t}_2\rangle_{\mathop{\rm Lie}} \subset \mathop{\rm Lie}(\mathop{\rm Aut}({\mathbb{A}}^2)).\] We assert that \({\mathfrak h}\) is not solvable and is not locally finite. Indeed, we have a decomposition \[\begin{align} \partial& = y^2\partial/\partial x-y(2x\partial/\partial x-y\partial/\partial y) +x(x\partial/\partial x-2y\partial/\partial y)+x^2\partial/\partial y\\ &= \frac{1}{3}(\partial_{-1,2} - 3\partial_{0,1}+3\partial_{1,0} -\partial_{2,-1}). \end{align}\] By Lemma 30, \({\mathfrak h}\) contains the locally nilpotent derivations \(y^2\partial/\partial x\) and \(x^2\partial/\partial y\), which generate a non-solvable, infinite-dimensional Lie subalgebra of \({\mathfrak h}\), see Lemma 37 or, alternatively, [2].
Lemma 39. Let \({\mathfrak h}\subset\mathop{\rm Lie}(\mathop{\rm Aut}({\mathbb{A}}^2))\) be a solvable Lie subalgebra of rank 2 generated by locally finite derivations. Then \({\mathfrak h}\) is triangulable.
Proof. By choosing appropriate coordinates in \({\mathbb{A}}^2\), we can assume that \({\mathfrak t}_2=\mathop{\rm Lie}({\mathbb{T}})\subset{\mathfrak h}\), see Remark 27. We state that, under this assumption, either \({\mathfrak h}\subset {\mathfrak j}_2^+\) or \({\mathfrak h}\subset {\mathfrak j}_2^-\).
Let us choose a non-algebraic semisimple derivation \(\delta=\delta_{\alpha,\beta}\in{\mathfrak t}_2\). By 2 , every homogeneous derivation \(\partial_{i,j}\) is an eigenvector of \(\delta\). By Lemma 30, for any \(\partial\in{\mathfrak h}\), all its homogeneous components \(c_{i,j}\partial_{i,j}\) belong to \({\mathfrak h}\).
Let \(\partial\notin {\mathfrak t}_2\) be locally finite. By Corollary 26, all vertices of the Newton polygon \(N(\partial)\) different from \((0,0)\) are Demazure points. According to Lemma 30, the corresponding homogeneous locally nilpotent derivations belong to \({\mathfrak h}\). By Lemma 37, all, except at most one, are of the same type, either of type \(\partial_{-1,k}\), or of type \(\partial_{l,-1}\). The same applies to the Newton polygons of all locally finite derivations in \({\mathfrak h}\). We can assume that all their vertices, except at most two, are of type \(\partial_{-1,k}\). The additional vertices can only be \((0,0)\) and \((0,-1)\). It follows that any such \(\partial\) is of the form \[\partial= p(y)\partial/\partial x+\alpha x\partial/\partial x +\beta y\partial/\partial y+\gamma \partial/\partial y, \quad\text{where}\quad p\in{\Bbbk}[y], \alpha, \beta,\gamma\in{\Bbbk},\] that is, \(\partial\in {\mathfrak j}^+_2\). Since this concerns all generators of \({\mathfrak h}\), we have \({\mathfrak h}\subset{\mathfrak j}^+_2\). ◻
The following corollary proves Theorem 19 in case where \(\mathop{\rm rk}({\mathfrak h})=2\).
Corollary 40. A solvable Lie subalgebra \({\mathfrak h}\subset\mathop{\rm Lie}(\mathop{\rm Aut}({\mathbb{A}}^2))\) of rank 2 generated by a finite set of locally finite derivations is triangulable and locally finite.
Proof. Given \(d\in{\mathbb{N}}\), consider the following algebraic subgroup of \({\mathop{\rm JONQ}}^+({\mathbb{A}}^2)\), \[{\mathop{\rm JONQ}}^+({\mathbb{A}}^2)_{\le d}= \{\phi\in{\mathop{\rm JONQ}}^+({\mathbb{A}}^2)\,|\, \mathop{\rm deg}(\phi)\le d\}\] and its Lie algebra \({\mathfrak j}^+_{2,\,\le d}= \mathop{\rm Lie}(\mathop{\rm JONQ}^+({\mathbb{A}}^2))_{\le d})\). By Lemma 39, \({\mathfrak h}\) is \(\mathop{\rm Ad}\)-conjugate to a Lie subalgebra \({\mathfrak h}'\subset {\mathfrak j}^+_{2,\,\le d}\), where \(d\) stands for the maximal degree of the generators of \({\mathfrak h}'\). Since \({\mathfrak h}'\subset{\mathfrak j}^+_{2,\,\le d}\) is locally finite, then \({\mathfrak h}\) is also locally finite. ◻
The following lemma concludes the proof of Theorem 19.
Lemma 41. Let \({\mathfrak h}\subset\mathop{\rm Lie}(\mathop{\rm Aut}({\mathbb{A}}^2))\) be a solvable Lie subalgebra generated by locally finite derivations \(\partial_1,\ldots,\partial_t\). Suppose that \({\mathfrak h}\) contains a semisimple derivation \(\delta\neq 0\). Then \({\mathfrak h}\) is locally finite and triangulable.
Proof. If \({\mathfrak h}\) contains an opportune pair, then it has rank 2 by Lemma 33. If \({\mathfrak h}\) is of rank 2, then the assertion follows from Corollary 40.
Suppose further that \({\mathfrak h}\) contains no opportune pair and is of rank \(1\). In this case, \(\delta\) is an algebraic derivation. Therefore, up to scaling and \(\mathop{\rm Ad}\)-conjugation, we can assume that \(\delta=\delta_{-m,n}\) for a pair \((n,m)\) of coprime integers, see Remark 27. By Corollary 35, \([\delta,\partial_j]=0\) for all \(j\). According to ?? –?? , we have \[\partial_{j}= \sum_i c_{j,i}\partial_{in, im},\] where the sum is finite. Thus, the Newton polygon \(N(\partial_j)\) is contained in the affine line \(L=\{-mx+ny=0\}\) passing through the origin. Furthermore, the endpoints of \(N(\partial_j)\) are Demazure points of type \((-1,l)\) and \((l',-1)\), see Corollary 26. This is only possible if each of these endpoints is either \((-1,1)\) or \((1,-1)\), and \((n,m)=\pm (1,-1)\). Therefore, \(\delta=\pm \partial_{0,0}=\pm (x\partial/\partial x-y\partial/\partial y)\), and \[{\mathfrak h}\subset {\mathfrak s}{\mathfrak l}_2=\langle \partial_{-1,1},\partial_{1,-1}\rangle_{\mathop{\rm Lie}} = \langle y\partial/\partial x, x\partial/\partial y\rangle_{\mathop{\rm Lie}}.\] Thus, \({\mathfrak h}\) is locally finite. Being solvable, \({\mathfrak h}\) is triangulable. ◻
The following corollary answers in affirmative, in the special case of the affine plane, a question communicated to the author by Andriy Regeta.
Corollary 42. Consider a Lie subalgebra \({\mathfrak h}\subset\mathop{\rm Lie}(\mathop{\rm Aut}({\mathbb{A}}^2))\). Suppose that \({\mathfrak h}\) is solvable and consisting of locally finite derivations. If \({\mathfrak h}\) is finitely generated, then it is locally finite. In general, \({\mathfrak h}=\varinjlim_i {\mathfrak h}_i\) is filtered by a sequence of locally finite Lie subalgebras. Furthermore, \({\mathfrak h}\) is triangulable.
Proof. First, suppose that \({\mathfrak h}\) consists of locally nilpotent derivations. Recall that the derived length of a solvable subalgebra of \(\mathop{\rm Lie}(\mathop{\rm Aut}({\mathbb{A}}^2))\) does not exceed \(4\), see [14]. It follows that the union of an increasing chain of solvable subalgebras of \(\mathop{\rm Lie}(\mathop{\rm Aut}({\mathbb{A}}^2))\) is again solvable. Therefore, by Zorn’s lemma, \({\mathfrak h}\) is contained in a maximal solvable subalgebra, say \({\mathfrak m}\) of \(\mathop{\rm Lie}(\mathop{\rm Aut}({\mathbb{A}}^2))\) consisting of locally nilpotent derivations.
Let \(\ker({\mathfrak m})=\cap \{\ker(\partial)\,|\,\partial\in{\mathfrak m}\}\). According to [15], if \(\ker({\mathfrak m})={\Bbbk}\), then \({\mathfrak m}\) is \(\mathop{\rm Ad}\)-conjugate to \({\mathfrak u}^+_2\), see Notation 20. Therefore, \({\mathfrak h}\) is triangulable, and thus filtered by a sequence of locally finite Lie subalgebras \({\mathfrak h}_{\le d}={\mathfrak h}\cap{\mathfrak u}^+_{2,\,\le d}\).
Otherwise, there is a nonconstant polynomial \(p\in\ker({\mathfrak m})\), such that every vector field \(\partial\in{\mathfrak m}\) is tangent to the fibers of \(p\colon{\mathbb{A}}^2\to{\mathbb{A}}^1\). Therefore, the general fiber of \(p\) is a disjoint union of \({\mathbb{A}}^1\)-curves. According to the Abhyankar–Moh–Suzuki theorem, \(p=q(u)\), where \(u\) is a variable of \({\Bbbk}[x,y]\) and \(q\in{\Bbbk}[u]\). In suitable coordinates \((u,v)\) on \({\mathbb{A}}^2\), every \(\partial\in{\mathfrak m}\) takes the Rentschler form \(\partial=r(u)\partial/\partial v\) with \(r\in{\Bbbk}[u]\). With respect to this coordinate system, \({\mathfrak m}\subset{\mathfrak u}^+_2\). Again, the above assertions apply to \({\mathfrak h}\).
Now suppose that \({\mathfrak h}\) contains a locally finite derivation \(\delta\neq 0\) that is not locally nilpotent.
If \({\mathfrak h}\) is finitely generated, then \({\mathfrak h}\) is locally finite and triangulable due to Theorem 4.
In general, \({\mathfrak h}\) is filtered by a sequence of finitely generated solvable Lie subalgebras. From the above, all of them are locally finite, and therefore integrable. Hence, \({\mathfrak h}\) is locally integrable in the terminology of [1]. By [1] Theorem 1.4 and the subsequent discussion, \({\mathfrak h}\) is contained in a maximal locally integrable Lie subalgebra \({\mathfrak b}\) of \(\mathop{\rm Lie}(\mathop{\rm Aut}({\mathbb{A}}^2))\), which is conjugate to \({\mathfrak j}_2^+\). Thus, \({\mathfrak h}\) is triangulable. ◻
In this subsection we prove Theorem 5. Let us recall this theorem.
Theorem 43. Let \({\mathfrak h}=\langle \partial_1,\ldots,\partial_k\rangle_{\mathop{\rm Lie}}\), where the \(\partial_i\in\mathop{\rm Vec}^{0}({\mathbb{A}}^2)\) are nonzero locally nilpotent derivations. Assume that the center of \({\mathfrak h}\) is nonzero. Then \({\mathfrak h}\) is locally finite and triangulable.
The proof is based on several auxiliary results. We begin with the following combinatorial lemma.
Lemma 44. Let \(\Pi_1\) and \(\Pi_2\) be two convex polygons in \({\mathbb{R}}^2\) and \(L\) a linear function on \({\mathbb{R}}^2\) such that \(L|_{\Pi_2}\) reaches its maximum at a single vertex \(v\) of \(\Pi_2\), and \(L|_{\Pi_1}\) reaches its maximum either at a single vertex \(u_1\) of \(\Pi_1\), or on a one-dimensional facet \(F_1=[u_1,u_2]\) of \(\Pi_1\). Consider the convex hull \(\Pi\) of \(\Pi_1+\Pi_2\). Then \(L|_{\Pi}\) attains its maximum at a single vertex \(u_1+v\) of \(\Pi\) in the first case, and on the one-dimensional facet \([u_1+v, u_2+v]\) of \(\Pi\) in the second case.
Proof. Suppose we are in the second case; the argument in the first case is similar. Let \(u_1,\ldots,u_l\) and \(v_1,\ldots,v_k\) be the vertices of \(\Pi_1\) and \(\Pi_2\), respectively, where \(v_1=v\). Then \(\Pi\) is the convex hull of the \(u_i+v_j\). We have \[\max(L|_{\Pi})={\max}_{i,j}\{L(u_i+v_j)\} ={\max}_i\{L(u_i)\} + {\max}_j\{L(v_j)\}= L(u_1+v)=L(u_2+v).\] Since \(L(u_i+v_j)<L(v+u_1)\) for \(i>2\) or \(j>1\), the lemma follows. ◻
We apply this lemma to determine the Newton polygon of the bracket of two derivations of \({\Bbbk}[x,y]\).
Lemma 45. Let \(\Pi_i=N(\partial_i)\), \(i=1,2\), where \(\partial_1,\partial_2\in\mathop{\rm Der}({\Bbbk}[x,y])\). Suppose that \(\Pi_1\) and \(\Pi_2\) admit a common linear function \(L\) satisfying the hypotheses of Lemma 44, where \(u_i=(k_i,l_i)\), \(i=1,2\), and \(v=(m,n)\). Assume that, for some \(i\in\{1,2\}\), we have \[\label{eq:ineqs} k_i+m, \, l_i+n \ge -1,\quad (k_i+m,\,l_i+n)\neq (-1,-1),\qquad{(5)}\] and \[\label{eq:non-vanish} [\partial_{m,n}, \partial_{k_i,l_i}]\neq 0, \quad\text{i.e.}\quad \det\left(\begin{matrix} k_i+1&m+1\\ l_i+1&n +1\end{matrix}\right) \neq 0\qquad{(6)}\] (see 1 ). Then \(u_i+v\) is a vertex of \(N([\partial_1,\partial_2])\). If, for \(i=1,2\), ?? and ?? are fulfilled, then \([u_1+v, u_2+v]\) is a facet of \(N([\partial_1,\partial_2])\).
Proof. Once again, suppose we are in the second case of Lemma 44; the argument in the first case is similar. According to Lemma 44, \(u_1+v\) and \(u_2+v\) are vertices of \(\Pi=N(\partial_1)+N(\partial_2)\) and \([u_1+v, u_2+v]\) is a facet of \(\Pi\). Let ?? and ?? be fulfilled for some \(i\in\{1,2\}\). Since \(N([\partial_1,\partial_2])\subset\Pi\), then \(u_i+v\) is a vertex of \(N([\partial_1,\partial_2])\). If ?? and ?? hold for both \(i=1\) and \(i=2\), then \([u_1+v, u_2+v]\) is a facet of \(N([\partial_1,\partial_2])\). ◻
Using Lemma ?? , we deduce the following results.
Lemma 46. Let \(\partial=p(y)\partial/ \partial x\), where \(p\) is a nonzero polynomial of degree \(n\), and let \(\delta\in\mathop{\rm Vec}^{0}({\mathbb{A}}^2)\setminus \{0\}\). Then \(\partial\) and \(\delta\) commute if and only if one of the following holds:
\(\delta=q(y)\partial/\partial x\) for some \(q\in{\Bbbk}[y]\);
\(\partial=c_0\partial/\partial x\) and \(\delta=c_1\partial/\partial y+q(y)\partial/\partial x\) for some \(c_0,c_1\in{\Bbbk}^*\) and some \(q\in{\Bbbk}[y]\).
Proof. It is evident that if either condition (i) or (ii) is satisfied, then \(\partial\) and \(\delta\) commute. To prove the converse, note that Newton’s polygon \(N(\partial)\) is either a vertical segment \([v_0,v]\), where \(v=(-1,n)\) and \(v_0=(-1,n_0)\) with \(n_0<n\), or a singleton \(\{v\}\). The linear function \(x\) restricted to \(N(\delta)\) attains its maximal value either at a single vertex, say \(u_1\), or on a facet \([u_1,u_2]\) of \(N(\delta)\), where for \(u_1=(a,b)\) we choose the upper end of this vertical segment. For \(\varepsilon>0\) sufficiently small, the linear function \(L=x+\varepsilon y\) reaches its maximal value \(a+\varepsilon b\) on \(N(\delta)\) at a single vertex \(u_1\). It attains its maximal value \(-1+\varepsilon n\) on \(N(\partial)\) at a single vertex \(v\).
If \([\partial_{-1,n},\partial_{a,b}]\neq 0\), then \(L\) attains its maximal value on the Newton polygon \(N([\partial,\delta])\) at a single vertex \(u_1+v\) of \(N([\partial,\delta])\), see Lemma 45. However, by hypotheses, \([\partial,\delta]=0\), and therefore \(N([\partial,\delta])\) is empty. This is only possible if \([\partial_{-1,n},\partial_{a,b}]= 0\).
Now, if \(a=\max\{x|_{N(\delta)}\}=-1\), then \(N(\delta)\) is contained in the vertical line \(x+1=0\), and therefore \(\delta=q(y)\partial/\partial x\). This corresponds to case (i).
Assume further that \(a\ge 0\). Then \([\partial_{-1,n},\partial_{a,b}]=0\) if and only if \(n=0\) and \((a,b)=(0,-1)\), see 1 . In this case, \(u_1=u_2=(0,-1)\). Thus, \(\partial=c_0\partial/\partial x\) and \(\delta=c_1\partial/\partial y+q(y)\partial/\partial x\). We are therefore in case (ii). ◻
Corollary 47. The centralizer of a locally nilpotent derivation \(\partial\) in \(\mathop{\rm Vec}^{0}({\mathbb{A}}^2)\) consists of locally nilpotent derivations and is either abelian or two-step nilpotent.
Proof. Indeed, by Rentschler’s theorem [16], in a suitable coordinate system on \({\mathbb{A}}^2\), \(\partial\) is of the form \(q(y)\partial/\partial x\). The result then follows from Lemma 46, since all \(\delta\) as in (i) and (ii) are locally nilpotent triangular derivations. ◻
Proof of Theorem 43. Choose a nonzero central element \(\partial\in{\mathfrak h}\). Since \([\partial,\partial_i]=0\), \(\partial\) is locally nilpotent, see Corollary 47. By Rentschler’s theorem [16], in a suitable coordinate system on \({\mathbb{A}}^2\), \(\partial\) is of the form \(q(y)\partial/\partial x\). According to Lemma 46, we have \(\partial_i\in{\mathfrak j}^+_2\) for \(i=1,\ldots,k\). Then, \({\mathfrak h}\subset{\mathfrak j}^+_{2,d}\) for some \(d>0\), cf. the proof of Corollary 40. The conclusion follows. ◻
Acknowledgments. We thank Hanspeter Kraft, who spotted an error in the first version of this article. It is also our pleasure to thank Ivan Arzhantsev for his interest and a useful remark, and Andriy Regeta for his question.